A Comparative Analysis of DTM and HAM
Solutions for Hunter-Saxton Equation and
Fisher Equation

DibyenduSaha; SanjibSengupta

doi:https://doi.org/10.14445/22315373/IJMTT-V56P534

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 56 | Number 4 | Year 2018 | Article Id. IJMTT-V56P534 | DOI : https://doi.org/10.14445/22315373/IJMTT-V56P534

A Comparative Analysis of DTM and HAM Solutions for Hunter-Saxton Equation and Fisher Equation

DibyenduSaha, SanjibSengupta

Citation :

DibyenduSaha, SanjibSengupta, "A Comparative Analysis of DTM and HAM Solutions for Hunter-Saxton Equation and Fisher Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 56, no. 4, pp. 236-243, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V56P534

Abstract

In this article, two partial differential equations (PDEs), namely Hunter-Saxton equation and Fisher equation, which are non-linear in nature and arise in different physical situations are adopted and their solutions compared by the differential transform method (DTM) and homotopy analysis method (HAM). The comparative analysis outlines the significant features and effectiveness of these methods to the nonlinear PDEs, as it is possible to find the closed form like approximate solutions with high degree of accuracy when compared to the exact solutions. Also, for the problems under consideration it has been observed that the HAM results better as compared to DTM. Beside both the methods discussed, can be extended to solve a class of problems arising in different practical situations.

Keywords

Nonlinear PDEs, Differential Transform Method (DTM), Homotopy Analysis Method (HAM), Hunter-Saxton Equation, Fisher Equation.

References

[1] J. K. Hunter, and R. Saxton, ―Dynamics of Director Fields,‖ Siam. J. Appl. Math., vol. 51(6), pp. 1498-1521, 1991.
[2] R. I. Ivanov, ―Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations,‖ Journal of Nonlinear Mathematical Physics, vol. 15, pp. 1-12, 2008.
[3] Z. Yin, ―On the Structure of Solutions to the Periodic Hunter-Saxton Equation,‖ Siam. J. Math. Anal., vol. 36(1), pp. 272-283, 2004.
[4] X. Wei, and Z. Yin, ―Global existence and blow-up phenomena for the periodic Hunter–Saxton equation with weak dissipation,‖ Journal of Nonlinear Mathematical Physics, vol. 18(1), pp. 139-149, 2011.
[5] P. Górka, and E. G. Reyes, ―The Modified Camassa–Holm Equation,‖ Int. Math. Res. Notices., rnq163, 2010.
[6] J. Lenells, ―The Hunter-Saxton Equation: A Geometric Approach,‖ Siam. J. Math. Anal., Vol. 40(1), pp. 266-277, 2008.
[7] P. Górka, and E. G. Reyes, ―The Modified Hunter–Saxton Equation,‖ J. Geom. Phys., vol. 62(8), pp. 1793-1809, 2012.
[8] E. G. Reyes, ―Pseudo-Potentials, Nonlocal Symmetries and Integrability of Some Shallow Water Equations,‖ Selecta Mathematica, vol. 12(2), pp. 241-270, 2006.
[9] F. Wang, F. Li, and Q. Chen, ―On the Cauchy problem for a weakly dissipative generalized $ $\mu $ $ μ-Hunter-Saxton equation,‖ MonatsheftefürMathematik, vol. 181(3), pp. 715-744, 2016.
[10] R. A. Fisher, ―The Wave of Advance of Advantageous Genes,‖ Ann. Eugenic., vol. 7(4), pp. 355-369, 1937.
[11] Y. Tan, H. Xu, and S. J. Liao, ―Explicit series solution of travelling waves with a front of Fisher equation,‖ Chaos, Solitons & Fractals, vol. 31(2), pp. 462-472, 2007.
[12] D. A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics, Princeton University Press, 2015.
[13] M. D. Bramson, ―Maximal Displacement of Branching Brownian Motion,‖ Commun. Pur. Appl. Math., vol. 31(5), pp. 531-581, 1978.
[14] H. C. Uckwell, Introduction to Theoretical Neurobiology: Volume 2, Nonlinear and Stochastic Theories Volume 8, Cambridge University Press, 2005.
[15] D. G. Aronson and H. F. Weinberger, Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation, In Partial Differential Equations and Related Topics, Berlin, Springer-Heidelberg, 1975.
[16] S. S. Behzadi, ―Numerical Solution of Hunter-Saxton Equation by Using Iterative Methods,‖ Journal of Informatics and Mathematical Sciences, vol. 3(2), pp. 127-143, 2011.
[17] Z. Yin, ―On the Structure of Solutions to the Periodic Hunter-Saxton Equation,‖ Siam. J. Math. Anal., vol. 36(1), pp. 272-283, 2004.
[18] M. T. Alquran, and K. Al-Khaled, ―Computation of eigenvalues and solutions for Sturm-Liouville problems using Sinc-Galerkin and differential transform methods,‖ Applications and Applied Mathematics: Applications and Applied Mathematics: An International Journal(AAM), vol. 5(1), pp. 128-47, 2010.